statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
iUnion_ball_nat (x : α) : ⋃ n : ℕ, ball x n = univ | iUnion_eq_univ_iff.2 fun y => exists_nat_gt (dist y x) | theorem | Metric.iUnion_ball_nat | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"exists_nat_gt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iUnion_ball_nat_succ (x : α) : ⋃ n : ℕ, ball x (n + 1) = univ | iUnion_eq_univ_iff.2 fun y => (exists_nat_gt (dist y x)).imp fun _ h => h.trans (lt_add_one _) | theorem | Metric.iUnion_ball_nat_succ | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"exists_nat_gt",
"lt_add_one"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall (x : α) (ε : ℝ) | { y | dist y x ≤ ε } | def | Metric.closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | `closedBall x ε` is the set of all points `y` with `dist y x ≤ ε` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_closedBall : y ∈ closedBall x ε ↔ dist y x ≤ ε | Iff.rfl | theorem | Metric.mem_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_closedBall' : y ∈ closedBall x ε ↔ dist x y ≤ ε | by rw [dist_comm, mem_closedBall] | theorem | Metric.mem_closedBall' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nonneg_of_mem_closedBall (hy : y ∈ closedBall x ε) : 0 ≤ ε | dist_nonneg.trans hy | theorem | Metric.nonneg_of_mem_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
sphere (x : α) (ε : ℝ) | { y | dist y x = ε } | def | Metric.sphere | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | `sphere x ε` is the set of all points `y` with `dist y x = ε` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_sphere : y ∈ sphere x ε ↔ dist y x = ε | Iff.rfl | theorem | Metric.mem_sphere | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_sphere' : y ∈ sphere x ε ↔ dist x y = ε | by rw [dist_comm, mem_sphere] | theorem | Metric.mem_sphere' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ne_of_mem_sphere (h : y ∈ sphere x ε) (hε : ε ≠ 0) : y ≠ x | ne_of_mem_of_not_mem h <| by simpa using hε.symm | theorem | Metric.ne_of_mem_sphere | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nonneg_of_mem_sphere (hy : y ∈ sphere x ε) : 0 ≤ ε | dist_nonneg.trans_eq hy | theorem | Metric.nonneg_of_mem_sphere | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
sphere_eq_empty_of_neg (hε : ε < 0) : sphere x ε = ∅ | Set.eq_empty_iff_forall_notMem.mpr fun _y hy => (nonneg_of_mem_sphere hy).not_gt hε | theorem | Metric.sphere_eq_empty_of_neg | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
sphere_eq_empty_of_subsingleton [Subsingleton α] (hε : ε ≠ 0) : sphere x ε = ∅ | Set.eq_empty_iff_forall_notMem.mpr fun _ h => ne_of_mem_sphere h hε (Subsingleton.elim _ _) | theorem | Metric.sphere_eq_empty_of_subsingleton | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
sphere_isEmpty_of_subsingleton [Subsingleton α] [NeZero ε] : IsEmpty (sphere x ε) | by
rw [sphere_eq_empty_of_subsingleton (NeZero.ne ε)]; infer_instance | instance | Metric.sphere_isEmpty_of_subsingleton | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"IsEmpty"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall_eq_singleton_of_subsingleton [Subsingleton α] (h : 0 ≤ ε) :
closedBall x ε = {x} | by
ext x'
simpa [Subsingleton.allEq x x'] | theorem | Metric.closedBall_eq_singleton_of_subsingleton | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_eq_singleton_of_subsingleton [Subsingleton α] (h : 0 < ε) : ball x ε = {x} | by
ext x'
simpa [Subsingleton.allEq x x'] | theorem | Metric.ball_eq_singleton_of_subsingleton | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_closedBall_self (h : 0 ≤ ε) : x ∈ closedBall x ε | by
rwa [mem_closedBall, dist_self] | theorem | Metric.mem_closedBall_self | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_self"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nonempty_closedBall : (closedBall x ε).Nonempty ↔ 0 ≤ ε | ⟨fun ⟨_x, hx⟩ => dist_nonneg.trans hx, fun h => ⟨x, mem_closedBall_self h⟩⟩ | theorem | Metric.nonempty_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall_eq_empty : closedBall x ε = ∅ ↔ ε < 0 | by
rw [← not_nonempty_iff_eq_empty, nonempty_closedBall, not_le]
@[simp] alias ⟨_, closedBall_of_neg⟩ := closedBall_eq_empty | theorem | Metric.closedBall_eq_empty | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"not_le"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall_eq_sphere_of_nonpos (hε : ε ≤ 0) : closedBall x ε = sphere x ε | Set.ext fun _ => (hε.trans dist_nonneg).ge_iff_eq' | theorem | Metric.closedBall_eq_sphere_of_nonpos | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Set.ext",
"dist_nonneg"
] | Closed balls and spheres coincide when the radius is non-positive | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ball_subset_closedBall : ball x ε ⊆ closedBall x ε | fun _y hy =>
mem_closedBall.2 (le_of_lt hy) | theorem | Metric.ball_subset_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"le_of_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
sphere_subset_closedBall : sphere x ε ⊆ closedBall x ε | fun _ => le_of_eq | theorem | Metric.sphere_subset_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"le_of_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
sphere_subset_ball {r R : ℝ} (h : r < R) : sphere x r ⊆ ball x R | fun _x hx ↦
(mem_sphere.1 hx).trans_lt h | lemma | Metric.sphere_subset_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall_disjoint_ball (h : δ + ε ≤ dist x y) : Disjoint (closedBall x δ) (ball y ε) | Set.disjoint_left.mpr fun _a ha1 ha2 =>
(h.trans <| dist_triangle_left _ _ _).not_gt <| add_lt_add_of_le_of_lt ha1 ha2 | theorem | Metric.closedBall_disjoint_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Disjoint",
"dist_triangle_left"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_disjoint_closedBall (h : δ + ε ≤ dist x y) : Disjoint (ball x δ) (closedBall y ε) | (closedBall_disjoint_ball <| by rwa [add_comm, dist_comm]).symm | theorem | Metric.ball_disjoint_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Disjoint",
"dist_comm",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_disjoint_ball (h : δ + ε ≤ dist x y) : Disjoint (ball x δ) (ball y ε) | (closedBall_disjoint_ball h).mono_left ball_subset_closedBall | theorem | Metric.ball_disjoint_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Disjoint"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall_disjoint_closedBall (h : δ + ε < dist x y) :
Disjoint (closedBall x δ) (closedBall y ε) | Set.disjoint_left.mpr fun _a ha1 ha2 =>
h.not_ge <| (dist_triangle_left _ _ _).trans <| add_le_add ha1 ha2 | theorem | Metric.closedBall_disjoint_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Disjoint",
"dist_triangle_left",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
sphere_disjoint_ball : Disjoint (sphere x ε) (ball x ε) | Set.disjoint_left.mpr fun _y hy₁ hy₂ => absurd hy₁ <| ne_of_lt hy₂ | theorem | Metric.sphere_disjoint_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Disjoint",
"ne_of_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_union_sphere : ball x ε ∪ sphere x ε = closedBall x ε | Set.ext fun _y => (@le_iff_lt_or_eq ℝ _ _ _).symm | theorem | Metric.ball_union_sphere | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Set.ext",
"le_iff_lt_or_eq",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
sphere_union_ball : sphere x ε ∪ ball x ε = closedBall x ε | by
rw [union_comm, ball_union_sphere] | theorem | Metric.sphere_union_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall_diff_sphere : closedBall x ε \ sphere x ε = ball x ε | by
rw [← ball_union_sphere, Set.union_diff_cancel_right sphere_disjoint_ball.symm.le_bot] | theorem | Metric.closedBall_diff_sphere | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Set.union_diff_cancel_right"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall_diff_ball : closedBall x ε \ ball x ε = sphere x ε | by
rw [← ball_union_sphere, Set.union_diff_cancel_left sphere_disjoint_ball.symm.le_bot] | theorem | Metric.closedBall_diff_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Set.union_diff_cancel_left"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_ball_comm : x ∈ ball y ε ↔ y ∈ ball x ε | by rw [mem_ball', mem_ball] | theorem | Metric.mem_ball_comm | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_closedBall_comm : x ∈ closedBall y ε ↔ y ∈ closedBall x ε | by
rw [mem_closedBall', mem_closedBall] | theorem | Metric.mem_closedBall_comm | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_sphere_comm : x ∈ sphere y ε ↔ y ∈ sphere x ε | by rw [mem_sphere', mem_sphere] | theorem | Metric.mem_sphere_comm | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_subset_ball (h : ε₁ ≤ ε₂) : ball x ε₁ ⊆ ball x ε₂ | fun _y yx =>
lt_of_lt_of_le (mem_ball.1 yx) h | theorem | Metric.ball_subset_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"lt_of_lt_of_le"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall_eq_bInter_ball : closedBall x ε = ⋂ δ > ε, ball x δ | by
ext y; rw [mem_closedBall, ← forall_gt_iff_le, mem_iInter₂]; rfl | theorem | Metric.closedBall_eq_bInter_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_subset_ball' (h : ε₁ + dist x y ≤ ε₂) : ball x ε₁ ⊆ ball y ε₂ | fun z hz =>
calc
dist z y ≤ dist z x + dist x y := dist_triangle _ _ _
_ < ε₁ + dist x y := by gcongr; exact hz
_ ≤ ε₂ := h | theorem | Metric.ball_subset_ball' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_triangle"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall_subset_closedBall (h : ε₁ ≤ ε₂) : closedBall x ε₁ ⊆ closedBall x ε₂ | fun _y (yx : _ ≤ ε₁) => le_trans yx h | theorem | Metric.closedBall_subset_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"le_trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall_subset_closedBall' (h : ε₁ + dist x y ≤ ε₂) :
closedBall x ε₁ ⊆ closedBall y ε₂ | fun z hz =>
calc
dist z y ≤ dist z x + dist x y := dist_triangle _ _ _
_ ≤ ε₁ + dist x y := by gcongr; exact hz
_ ≤ ε₂ := h | theorem | Metric.closedBall_subset_closedBall' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_triangle"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall_subset_ball (h : ε₁ < ε₂) : closedBall x ε₁ ⊆ ball x ε₂ | fun y (yh : dist y x ≤ ε₁) => lt_of_le_of_lt yh h | theorem | Metric.closedBall_subset_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"lt_of_le_of_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall_subset_ball' (h : ε₁ + dist x y < ε₂) :
closedBall x ε₁ ⊆ ball y ε₂ | fun z hz =>
calc
dist z y ≤ dist z x + dist x y := dist_triangle _ _ _
_ ≤ ε₁ + dist x y := by gcongr; exact hz
_ < ε₂ := h | theorem | Metric.closedBall_subset_ball' | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_triangle"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_le_add_of_nonempty_closedBall_inter_closedBall
(h : (closedBall x ε₁ ∩ closedBall y ε₂).Nonempty) : dist x y ≤ ε₁ + ε₂ | let ⟨z, hz⟩ := h
calc
dist x y ≤ dist z x + dist z y := dist_triangle_left _ _ _
_ ≤ ε₁ + ε₂ := add_le_add hz.1 hz.2 | theorem | Metric.dist_le_add_of_nonempty_closedBall_inter_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_triangle_left"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_lt_add_of_nonempty_closedBall_inter_ball (h : (closedBall x ε₁ ∩ ball y ε₂).Nonempty) :
dist x y < ε₁ + ε₂ | let ⟨z, hz⟩ := h
calc
dist x y ≤ dist z x + dist z y := dist_triangle_left _ _ _
_ < ε₁ + ε₂ := add_lt_add_of_le_of_lt hz.1 hz.2 | theorem | Metric.dist_lt_add_of_nonempty_closedBall_inter_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_triangle_left"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_lt_add_of_nonempty_ball_inter_closedBall (h : (ball x ε₁ ∩ closedBall y ε₂).Nonempty) :
dist x y < ε₁ + ε₂ | by
rw [inter_comm] at h
rw [add_comm, dist_comm]
exact dist_lt_add_of_nonempty_closedBall_inter_ball h | theorem | Metric.dist_lt_add_of_nonempty_ball_inter_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_lt_add_of_nonempty_ball_inter_ball (h : (ball x ε₁ ∩ ball y ε₂).Nonempty) :
dist x y < ε₁ + ε₂ | dist_lt_add_of_nonempty_closedBall_inter_ball <|
h.mono (inter_subset_inter ball_subset_closedBall Subset.rfl) | theorem | Metric.dist_lt_add_of_nonempty_ball_inter_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iUnion_closedBall_nat (x : α) : ⋃ n : ℕ, closedBall x n = univ | iUnion_eq_univ_iff.2 fun y => exists_nat_ge (dist y x) | theorem | Metric.iUnion_closedBall_nat | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"exists_nat_ge"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iUnion_inter_closedBall_nat (s : Set α) (x : α) : ⋃ n : ℕ, s ∩ closedBall x n = s | by
rw [← inter_iUnion, iUnion_closedBall_nat, inter_univ] | theorem | Metric.iUnion_inter_closedBall_nat | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_subset (h : dist x y ≤ ε₂ - ε₁) : ball x ε₁ ⊆ ball y ε₂ | fun z zx => by
rw [← add_sub_cancel ε₁ ε₂]
exact lt_of_le_of_lt (dist_triangle z x y) (add_lt_add_of_lt_of_le zx h) | theorem | Metric.ball_subset | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"dist_triangle",
"lt_of_le_of_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_half_subset (y) (h : y ∈ ball x (ε / 2)) : ball y (ε / 2) ⊆ ball x ε | ball_subset <| by rw [sub_self_div_two]; exact le_of_lt h | theorem | Metric.ball_half_subset | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"le_of_lt",
"sub_self_div_two"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_ball_subset_ball (h : y ∈ ball x ε) : ∃ ε' > 0, ball y ε' ⊆ ball x ε | ⟨_, sub_pos.2 h, ball_subset <| by rw [sub_sub_self]⟩ | theorem | Metric.exists_ball_subset_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
forall_of_forall_mem_closedBall (p : α → Prop) (x : α)
(H : ∃ᶠ R : ℝ in atTop, ∀ y ∈ closedBall x R, p y) (y : α) : p y | by
obtain ⟨R, hR, h⟩ : ∃ R ≥ dist y x, ∀ z : α, z ∈ closedBall x R → p z :=
frequently_iff.1 H (Ici_mem_atTop (dist y x))
exact h _ hR | theorem | Metric.forall_of_forall_mem_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | If a property holds for all points in closed balls of arbitrarily large radii, then it holds for
all points. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
forall_of_forall_mem_ball (p : α → Prop) (x : α)
(H : ∃ᶠ R : ℝ in atTop, ∀ y ∈ ball x R, p y) (y : α) : p y | by
obtain ⟨R, hR, h⟩ : ∃ R > dist y x, ∀ z : α, z ∈ ball x R → p z :=
frequently_iff.1 H (Ioi_mem_atTop (dist y x))
exact h _ hR | theorem | Metric.forall_of_forall_mem_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | If a property holds for all points in balls of arbitrarily large radii, then it holds for all
points. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isBounded_iff {s : Set α} :
IsBounded s ↔ ∃ C : ℝ, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C | by
rw [isBounded_def, ← Filter.mem_sets, @PseudoMetricSpace.cobounded_sets α, mem_setOf_eq,
compl_compl] | theorem | Metric.isBounded_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Filter.mem_sets",
"Set",
"compl_compl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
boundedSpace_iff : BoundedSpace α ↔ ∃ C, ∀ a b : α, dist a b ≤ C | by
rw [← isBounded_univ, Metric.isBounded_iff]
simp | lemma | Metric.boundedSpace_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"BoundedSpace",
"Metric.isBounded_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isBounded_iff_eventually {s : Set α} :
IsBounded s ↔ ∀ᶠ C in atTop, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C | isBounded_iff.trans
⟨fun ⟨C, h⟩ => eventually_atTop.2 ⟨C, fun _C' hC' _x hx _y hy => (h hx hy).trans hC'⟩,
Eventually.exists⟩ | theorem | Metric.isBounded_iff_eventually | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Set",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isBounded_iff_exists_ge {s : Set α} (c : ℝ) :
IsBounded s ↔ ∃ C, c ≤ C ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C | ⟨fun h => ((eventually_ge_atTop c).and (isBounded_iff_eventually.1 h)).exists, fun h =>
isBounded_iff.2 <| h.imp fun _ => And.right⟩ | theorem | Metric.isBounded_iff_exists_ge | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isBounded_iff_nndist {s : Set α} :
IsBounded s ↔ ∃ C : ℝ≥0, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → nndist x y ≤ C | by
simp only [isBounded_iff_exists_ge 0, NNReal.exists, ← NNReal.coe_le_coe, ← dist_nndist,
NNReal.coe_mk, exists_prop] | theorem | Metric.isBounded_iff_nndist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"NNReal.coe_le_coe",
"NNReal.coe_mk",
"Set",
"dist_nndist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
boundedSpace_iff_nndist : BoundedSpace α ↔ ∃ C, ∀ a b : α, nndist a b ≤ C | by
rw [← isBounded_univ, Metric.isBounded_iff_nndist]
simp | lemma | Metric.boundedSpace_iff_nndist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"BoundedSpace",
"Metric.isBounded_iff_nndist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
boundedSpace_iff_edist : BoundedSpace α ↔ ∃ C : ℝ≥0, ∀ a b : α, edist a b ≤ C | by
simp [Metric.boundedSpace_iff_nndist] | lemma | Metric.boundedSpace_iff_edist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"BoundedSpace",
"Metric.boundedSpace_iff_nndist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toUniformSpace_eq :
‹PseudoMetricSpace α›.toUniformSpace = .ofDist dist dist_self dist_comm dist_triangle | UniformSpace.ext PseudoMetricSpace.uniformity_dist | theorem | Metric.toUniformSpace_eq | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"UniformSpace.ext",
"dist_comm",
"dist_self",
"dist_triangle"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformity_basis_dist :
(𝓤 α).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { p : α × α | dist p.1 p.2 < ε } | by
rw [toUniformSpace_eq]
exact UniformSpace.hasBasis_ofFun (exists_gt _) _ _ _ _ _ | theorem | Metric.uniformity_basis_dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"UniformSpace.hasBasis_ofFun"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mk_uniformity_basis {β : Type*} {p : β → Prop} {f : β → ℝ}
(hf₀ : ∀ i, p i → 0 < f i) (hf : ∀ ⦃ε⦄, 0 < ε → ∃ i, p i ∧ f i ≤ ε) :
(𝓤 α).HasBasis p fun i => { p : α × α | dist p.1 p.2 < f i } | by
refine ⟨fun s => uniformity_basis_dist.mem_iff.trans ?_⟩
constructor
· rintro ⟨ε, ε₀, hε⟩
rcases hf ε₀ with ⟨i, hi, H⟩
exact ⟨i, hi, fun x (hx : _ < _) => hε <| lt_of_lt_of_le hx H⟩
· exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, H⟩ | theorem | Metric.mk_uniformity_basis | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"lt_of_lt_of_le"
] | Given `f : β → ℝ`, if `f` sends `{i | p i}` to a set of positive numbers
accumulating to zero, then `f i`-neighborhoods of the diagonal form a basis of `𝓤 α`.
For specific bases see `uniformity_basis_dist`, `uniformity_basis_dist_inv_nat_succ`,
and `uniformity_basis_dist_inv_nat_pos`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformity_basis_dist_rat :
(𝓤 α).HasBasis (fun r : ℚ => 0 < r) fun r => { p : α × α | dist p.1 p.2 < r } | Metric.mk_uniformity_basis (fun _ => Rat.cast_pos.2) fun _ε hε =>
let ⟨r, hr0, hrε⟩ := exists_rat_btwn hε
⟨r, Rat.cast_pos.1 hr0, hrε.le⟩ | theorem | Metric.uniformity_basis_dist_rat | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Metric.mk_uniformity_basis",
"exists_rat_btwn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformity_basis_dist_inv_nat_succ :
(𝓤 α).HasBasis (fun _ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 < 1 / (↑n + 1) } | Metric.mk_uniformity_basis (fun n _ => div_pos zero_lt_one <| Nat.cast_add_one_pos n) fun _ε ε0 =>
(exists_nat_one_div_lt ε0).imp fun _n hn => ⟨trivial, le_of_lt hn⟩ | theorem | Metric.uniformity_basis_dist_inv_nat_succ | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Metric.mk_uniformity_basis",
"Nat.cast_add_one_pos",
"div_pos",
"exists_nat_one_div_lt",
"le_of_lt",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformity_basis_dist_inv_nat_pos :
(𝓤 α).HasBasis (fun n : ℕ => 0 < n) fun n : ℕ => { p : α × α | dist p.1 p.2 < 1 / ↑n } | Metric.mk_uniformity_basis (fun _ hn => div_pos zero_lt_one <| Nat.cast_pos.2 hn) fun _ ε0 =>
let ⟨n, hn⟩ := exists_nat_one_div_lt ε0
⟨n + 1, Nat.succ_pos n, mod_cast hn.le⟩ | theorem | Metric.uniformity_basis_dist_inv_nat_pos | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Metric.mk_uniformity_basis",
"div_pos",
"exists_nat_one_div_lt",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformity_basis_dist_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
(𝓤 α).HasBasis (fun _ : ℕ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 < r ^ n } | Metric.mk_uniformity_basis (fun _ _ => pow_pos h0 _) fun _ε ε0 =>
let ⟨n, hn⟩ := exists_pow_lt_of_lt_one ε0 h1
⟨n, trivial, hn.le⟩ | theorem | Metric.uniformity_basis_dist_pow | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Metric.mk_uniformity_basis",
"exists_pow_lt_of_lt_one",
"pow_pos"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformity_basis_dist_lt {R : ℝ} (hR : 0 < R) :
(𝓤 α).HasBasis (fun r : ℝ => 0 < r ∧ r < R) fun r => { p : α × α | dist p.1 p.2 < r } | Metric.mk_uniformity_basis (fun _ => And.left) fun r hr =>
⟨min r (R / 2), ⟨lt_min hr (half_pos hR), min_lt_iff.2 <| Or.inr (half_lt_self hR)⟩,
min_le_left _ _⟩ | theorem | Metric.uniformity_basis_dist_lt | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Metric.mk_uniformity_basis",
"half_pos",
"min_le_left"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mk_uniformity_basis_le {β : Type*} {p : β → Prop} {f : β → ℝ}
(hf₀ : ∀ x, p x → 0 < f x) (hf : ∀ ε, 0 < ε → ∃ x, p x ∧ f x ≤ ε) :
(𝓤 α).HasBasis p fun x => { p : α × α | dist p.1 p.2 ≤ f x } | by
refine ⟨fun s => uniformity_basis_dist.mem_iff.trans ?_⟩
constructor
· rintro ⟨ε, ε₀, hε⟩
rcases exists_between ε₀ with ⟨ε', hε'⟩
rcases hf ε' hε'.1 with ⟨i, hi, H⟩
exact ⟨i, hi, fun x (hx : _ ≤ _) => hε <| lt_of_le_of_lt (le_trans hx H) hε'.2⟩
· exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, fun x (hx ... | theorem | Metric.mk_uniformity_basis_le | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"exists_between",
"le_trans",
"lt_of_le_of_lt"
] | Given `f : β → ℝ`, if `f` sends `{i | p i}` to a set of positive numbers
accumulating to zero, then closed neighborhoods of the diagonal of sizes `{f i | p i}`
form a basis of `𝓤 α`.
Currently we have only one specific basis `uniformity_basis_dist_le` based on this constructor.
More can be easily added if needed in t... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformity_basis_dist_le :
(𝓤 α).HasBasis ((0 : ℝ) < ·) fun ε => { p : α × α | dist p.1 p.2 ≤ ε } | Metric.mk_uniformity_basis_le (fun _ => id) fun ε ε₀ => ⟨ε, ε₀, le_refl ε⟩ | theorem | Metric.uniformity_basis_dist_le | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Metric.mk_uniformity_basis_le",
"le_refl"
] | Constant size closed neighborhoods of the diagonal form a basis
of the uniformity filter. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformity_basis_dist_le_inv_nat_succ :
(𝓤 α).HasBasis (fun _ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 ≤ 1 / (↑n + 1) } | Metric.mk_uniformity_basis_le (fun n _ => div_pos zero_lt_one <| Nat.cast_add_one_pos n)
fun _ε ε0 => (exists_nat_one_div_lt ε0).imp fun _n hn => ⟨trivial, hn.le⟩ | theorem | Metric.uniformity_basis_dist_le_inv_nat_succ | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Metric.mk_uniformity_basis_le",
"Nat.cast_add_one_pos",
"div_pos",
"exists_nat_one_div_lt",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformity_basis_dist_le_inv_nat_pos :
(𝓤 α).HasBasis (fun n : ℕ => 0 < n) fun n : ℕ => { p : α × α | dist p.1 p.2 ≤ 1 / ↑n } | Metric.mk_uniformity_basis_le (fun n hn => div_pos zero_lt_one <| Nat.cast_pos.2 hn) fun _ε ε0 =>
let ⟨n, hn⟩ := exists_nat_one_div_lt ε0
⟨n + 1, n.succ_pos, by simpa using hn.le⟩ | theorem | Metric.uniformity_basis_dist_le_inv_nat_pos | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Metric.mk_uniformity_basis_le",
"div_pos",
"exists_nat_one_div_lt",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformity_basis_dist_le_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
(𝓤 α).HasBasis (fun _ : ℕ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 ≤ r ^ n } | Metric.mk_uniformity_basis_le (fun _ _ => pow_pos h0 _) fun _ε ε0 =>
let ⟨n, hn⟩ := exists_pow_lt_of_lt_one ε0 h1
⟨n, trivial, hn.le⟩ | theorem | Metric.uniformity_basis_dist_le_pow | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Metric.mk_uniformity_basis_le",
"exists_pow_lt_of_lt_one",
"pow_pos"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_uniformity_dist {s : Set (α × α)} :
s ∈ 𝓤 α ↔ ∃ ε > 0, ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ s | uniformity_basis_dist.mem_uniformity_iff | theorem | Metric.mem_uniformity_dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_mem_uniformity {ε : ℝ} (ε0 : 0 < ε) : { p : α × α | dist p.1 p.2 < ε } ∈ 𝓤 α | mem_uniformity_dist.2 ⟨ε, ε0, fun _ _ ↦ id⟩ | theorem | Metric.dist_mem_uniformity | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | A constant size neighborhood of the diagonal is an entourage. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformContinuous_iff [PseudoMetricSpace β] {f : α → β} :
UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ ⦃a b : α⦄, dist a b < δ → dist (f a) (f b) < ε | uniformity_basis_dist.uniformContinuous_iff uniformity_basis_dist | theorem | Metric.uniformContinuous_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"PseudoMetricSpace",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
UniformContinuousOn f s ↔
∀ ε > 0, ∃ δ > 0, ∀ x ∈ s, ∀ y ∈ s, dist x y < δ → dist (f x) (f y) < ε | uniformity_basis_dist.uniformContinuousOn_iff uniformity_basis_dist | theorem | Metric.uniformContinuousOn_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"PseudoMetricSpace",
"Set",
"UniformContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_iff_le [PseudoMetricSpace β] {f : α → β} :
UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ ⦃a b : α⦄, dist a b ≤ δ → dist (f a) (f b) ≤ ε | uniformity_basis_dist_le.uniformContinuous_iff uniformity_basis_dist_le | theorem | Metric.uniformContinuous_iff_le | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"PseudoMetricSpace",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuousOn_iff_le [PseudoMetricSpace β] {f : α → β} {s : Set α} :
UniformContinuousOn f s ↔
∀ ε > 0, ∃ δ > 0, ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ δ → dist (f x) (f y) ≤ ε | uniformity_basis_dist_le.uniformContinuousOn_iff uniformity_basis_dist_le | theorem | Metric.uniformContinuousOn_iff_le | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"PseudoMetricSpace",
"Set",
"UniformContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_basis_ball : (𝓝 x).HasBasis (0 < ·) (ball x) | nhds_basis_uniformity uniformity_basis_dist | theorem | Metric.nhds_basis_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_nhds_iff : s ∈ 𝓝 x ↔ ∃ ε > 0, ball x ε ⊆ s | nhds_basis_ball.mem_iff | theorem | Metric.mem_nhds_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_nhds_iff {p : α → Prop} :
(∀ᶠ y in 𝓝 x, p y) ↔ ∃ ε > 0, ∀ ⦃y⦄, dist y x < ε → p y | mem_nhds_iff | theorem | Metric.eventually_nhds_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"eventually_nhds_iff",
"mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_nhds_iff_ball {p : α → Prop} :
(∀ᶠ y in 𝓝 x, p y) ↔ ∃ ε > 0, ∀ y ∈ ball x ε, p y | mem_nhds_iff | theorem | Metric.eventually_nhds_iff_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_nhds_prod_iff {f : Filter ι} {x₀ : α} {p : α × ι → Prop} :
(∀ᶠ x in 𝓝 x₀ ×ˢ f, p x) ↔ ∃ ε > (0 : ℝ), ∃ pa : ι → Prop, (∀ᶠ i in f, pa i) ∧
∀ ⦃x⦄, dist x x₀ < ε → ∀ ⦃i⦄, pa i → p (x, i) | by
refine (nhds_basis_ball.prod f.basis_sets).eventually_iff.trans ?_
simp only [Prod.exists, forall_prod_set, id, mem_ball, and_assoc, exists_and_left,
Set.mem_surjective.exists, eventually_mem_set] | theorem | Metric.eventually_nhds_prod_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Filter"
] | A version of `Filter.eventually_prod_iff` where the first filter consists of neighborhoods
in a pseudo-metric space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
eventually_prod_nhds_iff {f : Filter ι} {x₀ : α} {p : ι × α → Prop} :
(∀ᶠ x in f ×ˢ 𝓝 x₀, p x) ↔ ∃ pa : ι → Prop, (∀ᶠ i in f, pa i) ∧
∃ ε > 0, ∀ ⦃i⦄, pa i → ∀ ⦃x⦄, dist x x₀ < ε → p (i, x) | by
rw [eventually_swap_iff, Metric.eventually_nhds_prod_iff]
constructor <;>
· rintro ⟨a1, a2, a3, a4, a5⟩
exact ⟨a3, a4, a1, a2, fun _ b1 b2 b3 => a5 b3 b1⟩ | theorem | Metric.eventually_prod_nhds_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Filter",
"Metric.eventually_nhds_prod_iff"
] | A version of `Filter.eventually_prod_iff` where the second filter consists of neighborhoods
in a pseudo-metric space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nhds_basis_closedBall : (𝓝 x).HasBasis (fun ε : ℝ => 0 < ε) (closedBall x) | nhds_basis_uniformity uniformity_basis_dist_le | theorem | Metric.nhds_basis_closedBall | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_basis_ball_inv_nat_succ :
(𝓝 x).HasBasis (fun _ => True) fun n : ℕ => ball x (1 / (↑n + 1)) | nhds_basis_uniformity uniformity_basis_dist_inv_nat_succ | theorem | Metric.nhds_basis_ball_inv_nat_succ | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_basis_ball_inv_nat_pos :
(𝓝 x).HasBasis (fun n => 0 < n) fun n : ℕ => ball x (1 / ↑n) | nhds_basis_uniformity uniformity_basis_dist_inv_nat_pos | theorem | Metric.nhds_basis_ball_inv_nat_pos | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_basis_closedBall_inv_nat_succ :
(𝓝 x).HasBasis (fun _ => True) fun n : ℕ => closedBall x (1 / (↑n + 1)) | nhds_basis_uniformity uniformity_basis_dist_le_inv_nat_succ | theorem | Metric.nhds_basis_closedBall_inv_nat_succ | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_basis_closedBall_inv_nat_pos :
(𝓝 x).HasBasis (fun n => 0 < n) fun n : ℕ => closedBall x (1 / ↑n) | nhds_basis_uniformity uniformity_basis_dist_le_inv_nat_pos | theorem | Metric.nhds_basis_closedBall_inv_nat_pos | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_basis_ball_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
(𝓝 x).HasBasis (fun _ => True) fun n : ℕ => ball x (r ^ n) | nhds_basis_uniformity (uniformity_basis_dist_pow h0 h1) | theorem | Metric.nhds_basis_ball_pow | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_basis_closedBall_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
(𝓝 x).HasBasis (fun _ => True) fun n : ℕ => closedBall x (r ^ n) | nhds_basis_uniformity (uniformity_basis_dist_le_pow h0 h1) | theorem | Metric.nhds_basis_closedBall_pow | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_iff : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ball x ε ⊆ s | by
simp only [isOpen_iff_mem_nhds, mem_nhds_iff] | theorem | Metric.isOpen_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"IsOpen",
"isOpen_iff_mem_nhds",
"mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_ball : IsOpen (ball x ε) | isOpen_iff.2 fun _ => exists_ball_subset_ball | theorem | Metric.isOpen_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"IsOpen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : ball x ε ∈ 𝓝 x | isOpen_ball.mem_nhds (mem_ball_self ε0) | theorem | Metric.ball_mem_nhds | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : closedBall x ε ∈ 𝓝 x | mem_of_superset (ball_mem_nhds x ε0) ball_subset_closedBall | theorem | Metric.closedBall_mem_nhds | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closedBall_mem_nhds_of_mem {x c : α} {ε : ℝ} (h : x ∈ ball c ε) : closedBall c ε ∈ 𝓝 x | mem_of_superset (isOpen_ball.mem_nhds h) ball_subset_closedBall | theorem | Metric.closedBall_mem_nhds_of_mem | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithin_basis_ball {s : Set α} :
(𝓝[s] x).HasBasis (fun ε : ℝ => 0 < ε) fun ε => ball x ε ∩ s | nhdsWithin_hasBasis nhds_basis_ball s | theorem | Metric.nhdsWithin_basis_ball | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Set",
"nhdsWithin_hasBasis"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_nhdsWithin_iff {t : Set α} : s ∈ 𝓝[t] x ↔ ∃ ε > 0, ball x ε ∩ t ⊆ s | nhdsWithin_basis_ball.mem_iff | theorem | Metric.mem_nhdsWithin_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_nhdsWithin_nhdsWithin [PseudoMetricSpace β] {t : Set β} {f : α → β} {a b} :
Tendsto f (𝓝[s] a) (𝓝[t] b) ↔
∀ ε > 0, ∃ δ > 0, ∀ ⦃x : α⦄, x ∈ s → dist x a < δ → f x ∈ t ∧ dist (f x) b < ε | (nhdsWithin_basis_ball.tendsto_iff nhdsWithin_basis_ball).trans <| by
simp only [inter_comm _ s, inter_comm _ t, mem_inter_iff, and_imp, gt_iff_lt, mem_ball] | theorem | Metric.tendsto_nhdsWithin_nhdsWithin | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | [] | [
"PseudoMetricSpace",
"Set",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.